Trigonometry Examples

$5 \tan (B) + \sqrt{13} = 0$

Step 1

Subtract $\sqrt{13}$ from both sides of the equation.

$5 \tan (B) = - \sqrt{13}$

Step 2

Divide each term in $5 \tan (B) = - \sqrt{13}$ by $5$ and simplify.

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Step 2.1

Divide each term in $5 \tan (B) = - \sqrt{13}$ by $5$ .

$\frac{5 \tan (B)}{5} = \frac{- \sqrt{13}}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{5 \tan (B)}{5} = \frac{- \sqrt{13}}{5}$

Step 2.2.1.2

Divide $\tan (B)$ by $1$ .

$\tan (B) = \frac{- \sqrt{13}}{5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$\tan (B) = - \frac{\sqrt{13}}{5}$

Step 3

Take the inverse tangent of both sides of the equation to extract $B$ from inside the tangent.

$B = \arctan (- \frac{\sqrt{13}}{5})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arctan (- \frac{\sqrt{13}}{5})$ .

$B = - 35.79575991$

Step 5

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the third quadrant.

$B = - 35.79575991 - 180$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Add $360 °$ to $- 35.79575991 - 180 °$ .

$B = - 35.79575991 - 180 ° + 360 °$

Step 6.2

The resulting angle of $144.20424008 °$ is positive and coterminal with $- 35.79575991 - 180$ .

$B = 144.20424008 °$

Step 7

Find the period of $\tan (B)$ .

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Step 7.1

The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

Step 7.2

Replace $b$ with $1$ in the formula for period.

$\frac{180}{| 1 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{180}{1}$

Step 7.4

Divide $180$ by $1$ .

$180$

Step 8

Add $180$ to every negative angle to get positive angles.

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Step 8.1

Add $180$ to $- 35.79575991$ to find the positive angle.

$- 35.79575991 + 180$

Step 8.2

Subtract $35.79575991$ from $180$ .

$144.20424008$

Step 8.3

List the new angles.

$B = 144.20424008$

Step 9

The period of the $\tan (B)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$B = 144.20424008 + 180 n, 144.20424008 + 180 n$ , for any integer $n$

Step 10

Consolidate $144.20424008 + 180 n$ and $144.20424008 + 180 n$ to $144.20424008 + 180 n$ .

$B = 144.20424008 + 180 n$ , for any integer $n$